Generalized gamma distribution

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Description

Density, distribution function, hazards, quantile function and random generation for the generalized gamma distribution, using the parameterisation originating from Prentice (1974). Also known as the (generalized) log-gamma distribution.

Usage

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     dgengamma(x, mu=0, sigma=1, Q, log = FALSE)
     pgengamma(q, mu=0, sigma=1, Q, lower.tail = TRUE, log.p = FALSE)
     qgengamma(p, mu=0, sigma=1, Q, lower.tail = TRUE, log.p = FALSE)
     rgengamma(n, mu=0, sigma=1, Q)
     Hgengamma(x, mu=0, sigma=1, Q)
     hgengamma(x, mu=0, sigma=1, Q)

Arguments

x,q

vector of quantiles.

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

mu

Vector of “location” parameters.

sigma

Vector of “scale” parameters. Note the inconsistent meanings of the term “scale” - this parameter is analogous to the (log-scale) standard deviation of the log-normal distribution, “sdlog” in dlnorm, rather than the “scale” parameter of the gamma distribution dgamma. Constrained to be positive.

Q

Vector of shape parameters.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are P(X <= x), otherwise, P(X > x).

Details

If g ~ Gamma(Q^{-2}, 1) , and w = log(Q^2*g) / Q, then x = exp(mu + sigma w) follows the generalized gamma distribution with probability density function

f(x | mu, sigma, Q) = |Q| (Q^{-2})^{Q^{-2}} / / (sigma * x * Gamma(Q^{-2})) exp(Q^{-2}*(Q*w - exp(Q*w)))

This parameterisation is preferred to the original parameterisation of the generalized gamma by Stacy (1962) since it is more numerically stable near to Q=0 (the log-normal distribution), and allows Q<=0. The original is available in this package as dgengamma.orig, for the sake of completion and compatibility with other software - this is implicitly restricted to Q>0 (or k>0 in the original notation). The parameters of dgengamma and dgengamma.orig are related as follows.

dgengamma.orig(x, shape=shape, scale=scale, k=k) =

dgengamma(x, mu=log(scale) + log(k)/shape, sigma=1/(shape*sqrt(k)), Q=1/sqrt(k))

The generalized gamma distribution simplifies to the gamma, log-normal and Weibull distributions with the following parameterisations:

dgengamma(x, mu, sigma, Q=0) = dlnorm(x, mu, sigma)
dgengamma(x, mu, sigma, Q=1) = dweibull(x, shape=1/sigma, scale=exp(mu))
dgengamma(x, mu, sigma, Q=sigma) = dgamma(x, shape=1/sigma^2, rate=exp(-mu) / sigma^2)

The properties of the generalized gamma and its applications to survival analysis are discussed in detail by Cox (2007).

The generalized F distribution GenF extends the generalized gamma to four parameters.

Value

dgengamma gives the density, pgengamma gives the distribution function, qgengamma gives the quantile function, rgengamma generates random deviates, Hgengamma retuns the cumulative hazard and hgengamma the hazard.

Author(s)

Christopher Jackson <chris.jackson@mrc-bsu.cam.ac.uk>

References

Prentice, R. L. (1974). A log gamma model and its maximum likelihood estimation. Biometrika 61(3):539-544.

Farewell, V. T. and Prentice, R. L. (1977). A study of distributional shape in life testing. Technometrics 19(1):69-75.

Lawless, J. F. (1980). Inference in the generalized gamma and log gamma distributions. Technometrics 22(3):409-419.

Cox, C., Chu, H., Schneider, M. F. and Muñoz, A. (2007). Parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution. Statistics in Medicine 26:4252-4374

Stacy, E. W. (1962). A generalization of the gamma distribution. Annals of Mathematical Statistics 33:1187-92

See Also

GenGamma.orig, GenF, Lognormal, GammaDist, Weibull.